Integrals over General Regions¶
The next step is to consider the volume underneath a surface $\Gamma(f,D)$, where $D$ is a general region on the plane that is not necessarily a rectangle. We begin by briefly describing the problem.
Video lecture
Type I Regions¶
A region $D \subseteq \mathbb{R}^2$ is said to be of type I if $D$ is bounded between two functions $y = g_1(x)$ and $y = g_2(x)$ in the $y$-direction, and $D$ is bounded between $x = a$ and $x = b$ in the $x$-direction. Therefore
$$
\int\!\!\!\int_D f(x,y)\, dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\, dy\, dx.
$$
Type II Regions¶
A region $D \subseteq \mathbb{R}^2$ is said to be of type II if $D$ is bounded between two functions $x = h_1(y)$ and $x = h_2(y)$ in the $x$-direction, and $D$ is bounded between $y = c$ and $y = d$ in the $y$-direction. Therefore
$$
\int\!\!\!\int_D f(x,y)\, dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y)\, dx\, dy.
$$
In case $D\subseteq \mathbb{R}^2$ is neither type I nor type II, then setting up the double integral may require more care. As discussed in the video lecture above, it may be possible to divide the region into sub-regions that are either type I or type II. Other times, we might need to transform the region entirely. We'll think about the latter idea again in section 15.9.
1. Example¶
Compute the double integral.
$$
\int_0^{\frac{\pi}{2}} \int_0^x x\sin(y)\, dy\, dx
$$
Check your answer
$$\frac{\pi^2}{8} - \frac{\pi}{2} + 1$$
Video solution
2. Example¶
Compute the double integral,
$$
\int\!\!\!\int_D \frac{x}{y^2 + 1}\, dA,
$$
where $D$ is the region defined by
$$D = \big\{ (x,y) \mid 0 \leq x \leq \sqrt{y}\,,\ \ 0\leq y \leq 4 \big\}.$$
Check your answer
$$\ln\,17^{\tfrac{1}{4}}$$
Video solution
3. Example¶
Compute the double integral
$$
\int\!\!\!\int_D y^2e^{xy}\, dA,
$$
where $D$ is the region bounded between the curves $y=x$, $y = 4$, and $x = 0$.
This region could be treated as either type I or type II. Take some time at the beginning of this example to carefully think about which point-of-view would be easier for this problem.
Check your answer
$$\frac{1}{2}\Big( e^{16} - 17\Big)$$
Video solution
Discussion¶
Questions? You can ask them here.